Spatial Dynamic Panel Model and System GMM: A Monte Carlo Investigation

Transcription

1 Spatial Dynamic Panel Model and System GMM: A Monte Carlo Investigation Madina Kukenova y José-Antonio Monteiro z Draft: May 2009 Abstract This paper investigates the nite sample properties of estimators for spatial dynamic panel models in the presence of several endogenous variables. So far, none of the available estimators in spatial econometrics allows considering spatial dynamic models with one or more endogenous variables. We propose to apply system-gmm, since it can correct for the endogeneity of the dependent variable, the spatial lag as well as other potentially endogenous variables using internal and/or external instruments. The Monte-Carlo investigation compares the performance of spatial MLE, spatial dynamic MLE (Elhorst (2005)), spatial dynamic QMLE (Yu et al. (2008)), LSDV, di erence-gmm (Arellano & Bond (1991)), as well as extended- GMM (Arellano & Bover (1995), Blundell & Bover (1998)) in terms of bias and root mean squared error. The results suggest that, in order to account for the endogeneity of several covariates, spatial dynamic panel models should be estimated using extended GMM. On a practical ground, this is also important, because system- GMM avoids the inversion of high dimension spatial weights matrices, which can be computationally demanding for large N and/or T. Keywords: Spatial Econometrics, Dynamic Panel Model, System GMM, Maximum Likelihood, Monte Carlo Simulations JEL classi cation: C15, C31, C33 We thank Florian Pelgrin for helpful comments. y University of Lausanne, madina.kukenova@unil.ch z University of Neuchatel, jose-antonio.monteiro@unine.ch 1

2 1 Introduction Although the econometric analysis of dynamic panel models has drawn a lot of attention in the last decade, econometric analysis of spatial and dynamic panel models is almost inexistent. So far, none of the available estimators allows to consider a dynamic spatial lag panel model with one or more endogenous variables (besides the time and spatial lag) as explanatory variables. Empirically, there are numerous examples where the presence of a dynamic process, spatial dependence and endogeneity might occur. This is the case with the analysis of the determinants of complex Foreign Direct Investment (FDI). Complex FDI is undertaken by a multinational rm from home country i investing in production plants in several host countries to explore the comparative advantages of these locations (Baltagi et al. (2007)). This type of FDI can, thus, feature complementary/substitutive spatial dependence with respect to FDI to other host countries. The presence of complex FDI can be tested empirically by estimating a spatial lag model (as proposed by Blonigen et al. (2007)), which can also include a lagged dependent variable to account for the fact that FDI decisions are part of a dynamic process, i.e. more FDI in a host country seems to attract more FDI in this same host country. Additional endogenous variables like environmental stringency, tari s or taxes can be included. These variables in uence the decision of multinational to invest in a country but are also determined or in uenced by the strategic interaction of these multinationals The inclusion of the lagged dependent variable and additional endogenous variables in a spatial lag model invalidate the use of traditional spatial maximum likelihood estimators, that is why this model is estimated in several empirical studies (Kukenova & Monteiro (2009), Madariaga & Poncet (2007)) using the system generalized method of moments (GMM) estimator, developed by Arellano & Bover (1995) and Blundell & Bond (1998). The main argument of applying the extended GMM in a spatial context is that it corrects for the endogeneity of the spatial lagged dependent variable and other potentially endogenous explanatory variables. In addition, extended GMM is robust to some econometrics issues such as measurement error and weak instruments. It can also control for arbitrary heteroskedasticity and autocorrelation in the disturbance error. Its implementation is computationally tractable as it avoids the inversion of high dimension spatial weights matrix W and the computation of its eigenvalues. 2

3 Going beyond this intuitive motivation, we conduct an extensive Monte Carlo study comparing performance of extended GMM with performance of several spatial estimators (Spatial MLE (SMLE), Spatial Dynamic MLE (SDMLE) and Spatial Dynamic QMLE (SDQMLE)) in terms of bias and root mean squared error criteria. We verify the sensitivity of results to alternative distributional assumptions and alternative speci cation of the matrices of spatial weights. Finally, we check robustness of the estimators to misspeci cation of the matrices of spatial weights, which is an important issue in spatial econometrics. The outline of the paper is as follows. The dynamic spatial lag model is de ned in section 2. The Monte Carlo investigation is described and performed in section 3. Section 4 checks the robustness of the main results. Finally, section 5 concludes. 2 Spatial Dynamic Panel Model Spatial data is characterized by the spatial arrangement of the observations. Following Tobler s First Law of Geography, everything is related to everything else, but near things are more related than distant things, the spatial linkages of the observations i = 1; :::; N are measured by de ning a spatial weight matrix 1, denoted by W t for any year t = 1; :::; T : w t (d k;j ) w t (d k;l ) w W t = t (d j;k ) 0 w t (d j;l ). B..... A w t (d l;k ) w t (d l;j ) 0 where w t (d j;k ) de nes the functional form of the weights between any two pair of location j and k. In the construction of the weights themselves, the theoretical foundation for w t (d j;k ) is quite general and the particular functional form of any single element in W t is, therefore, not prescribed. In fact, the determination of the proper speci cation of W t is one of the most di cult and controversial methodological issues in spatial data analysis. In empirical applications, geographical distance or economic distance are usually used. 1 The modelling of spatial dependence can either be done with lattice models (i.e. weight matrix (Anselin (1988)) or geostatistical models (i.e. function of separative distances (Chen & Conley (2001)). 3

4 As is standard in spatial econometrics, for ease of interpretation, the weighting matrix W t is row standardized so that each row in W t sums to one. As distances are timeinvariant, it will generally be the case that W t = W t+1. However, when dealing with unbalanced panel data, this is no longer true. Stacking the data rst by time and then by cross-section, the full weighting matrix, W, is given by: 0 1 W W =. B A 0 0 W T The development of empirical spatial models is intimately linked to the recent progress in spatial econometrics. The basic spatial model was suggested by Cli & Ord (1981), but it did not receive important theoretical extensions until the middle of the 1990s. Anselin (2001, 2006), Elhorst (2003b) & Lee & Yu (2009) provide thorough surveys of the di erent spatial models and suggest econometric strategies to estimate them. 2.1 Dynamic Spatial Lag Model A general spatial dynamic panel model, also known as a spatial dynamic autoregressive model with spatial autoregressive error (SARAR(1,1)), can be described as follows: Y t = Y t 1 + W 1t Y t + EX t + EN t + " t (1) " t = + W 2t " t + v t ; t = 1; :::; T where Y t is a N 1 vector, W 1t and W 2t are N N spatial weight matrices which are non-stochastic and exogenous to the model, is the vector of country e ect, EX t is a N p matrix of p exogenous explanatory variables (p 0) and EN t is a N q matrix of q endogenous explanatory variables with respect to Y t (q 0). Finally, v t is assumed to be distributed as (0; ). By adding some restrictions to the parameters, two popular spatial model speci cations can be derived from this general spatial model, namely the dynamic spatial lag model ( = 0) and the dynamic spatial error model ( = 0) 2. 2 The analysis of the spatial error panel model and the spatial lag with spatial error model is beyond the scope of this paper. For further details, see Elhorst (2005), Kapoor et al. (2007), Kelejian & Prucha (2007), Mutl & Pfa ermeier (2008) as well as Lee & Ju (2009). 4

5 The spatial lag model accounts directly for relationships between dependent variables that are believed to be related in some spatial way. Somewhat analogous to a lagged dependent variable in time series analysis, the estimated spatial lag coef- cient 3 characterizes the contemporaneous correlation between one cross-section and other geographically-proximate cross-sections. The following equation gives the basic spatial dynamic panel speci cation, also known as the "time-space simultaneous" model (Anselin (1988, 2001)) 4 : Y t = Y t 1 + W t Y t + EX t + EN t + + v t (2) The spatial autoregressive coe cient () associated with W t Y t represents the e ect of the weighted average (w t (d ij ) being the weights) of the neighborhood, i.e. [W t Y t ] i = P j=1::n t w t (d ij )Y jt. The spatial lag term allows to determine if the dependent variable Y t is (positively/negatively) a ected by the Y t from other close locations weighted by a given criterion (usually distance or contiguity). In other words, the spatial lag coe cient captures the impact of Y t from neighborhood locations. Let! min and! max be the smallest and highest characteristic root of the spatial matrix W, then this spatial e ect is assumed to lie between 1! min and 1! max. Most of the spatial econometrics literature constrains the spatial lag to lie between -1 and +1. However, this might be restrictive, because if the spatial matrix is row-normalized, then the highest characteristic root is equal to unity (! max = 1), but the smallest eigenvalue can be bigger than -1, which would lead the lower bound to be smaller than -1. Given that expression (2) is a combination of a time and spatial autoregressive models, we need to ensure that the resulting process is stationary. The stationarity restrictions in this model are stronger than the individual restrictions imposed on the coe cients of a spatial or dynamic model. The process is covariance stationary if (I N W t ) 1 < 1, or, equivalently, if jj < 1! max if 0 jj < 1! min if < 0 3 The spatial autoregressive term is also referred as endogenous interation e ects in social economics or as interdependence process in political science. 4 Beside the "time-space simulatenous" model, Anselin distinguishes three other distinct spatial lag panel models: the "pure space recursive" model which only includes a lagged spatial lag coe cient; the "time-space recursive" speci cation which considers a lagged dependent variable as well as a lagged spatial lag (see Korniotis (2007)); and the "time-space dynamic" model, which includes a time lag, a spatial lag and a lagged spatial lag. 5

6 From an econometric viewpoint, equation (2) faces simultaneity and endogeneity problems, which in turn means that OLS estimation will be biased and inconsistent (Anselin (1988)). To see this point more formally, note that the reduced form of equation(2) takes the following form: Y t = (I N W t ) 1 (Y t 1 + EX t + EN t + + v t ) Each element of Y t is a linear combination of all of the error terms. Moreover, as pointed out by Anselin (2003), assuming jj < 1 and each element of W t is smaller than one imply that (I N W t ) 1 can be reformulated as a Leontief expansion (I N W t ) 1 = I + W t + 2 Wt 2 +:::: Accordingly, the spatial lag model features two types of global spillovers e ects: a multiplier e ect for the predictor variables as well as a di usion e ect for the error process. Since the spatial lag term W t Y t is correlated with the disturbances, even if v t are independently and identically distributed, it must be treated as an endogenous variable and proper estimation method must account for this endogeneity. Despite the fact that dynamic panel models have been the object of recent important developments (see the survey by Baltagi & Kao (2000) or Phillips & Moon (2000)), econometric analysis of spatial dynamic panel models is almost inexistent. In fact, there is only a limited number of available estimators that deal with spatial and time dependence in a panel setting. Table 1 summarizes the di erent estimators proposed in the literature. In the absence of spatial dependence, there are three main types of estimators available to estimate a dynamic panel model. The rst type of estimators consists of estimating an unconditional likelihood function (Hsiao et al. (2002)). The second type of procedure corrects the bias associated with the least square dummy variables (LSDV) estimator (Bun & Carree (2005)). The last type, which is the most popular, relies on GMM estimators, like di erence GMM (Arellano & Bond (1992)), system GMM (Arellano & Bover (1995), Blundell & Bond (1998)) or continuously updated GMM (Hansen et al. (1996)). Assuming all explanatory variables are exogenous beside the spatial autoregressive term, the spatial lag panel model without any time dynamic is usually estimated using spatial maximum likelihood (Elhorst (2003b)) or spatial two-stage least squares methods (S2SLS) (Anselin (1988) (2001)). The ML approach consists of estimating the spatial coe cient by maximizing the non-linear reduced form of the spatial lag model. 6

7 Table 1: Spatial Dynamic Estimators Survey Model Estimation Methods Endogenous Yt= Y t 1+EX t +t GMM (Arellano et al. (1991) (1995), Blundell et al. (1998), Hansen et al. (1996)) Yt 1; MLE/Minimum Distance (Hsiao, Pesaran & Tahmiscioglu (2002)) CLSDV (Kiviet (1995), Hahn & Kuersteiner (2002) Bun & Carree (2005)) Yt= Y t 1+EX t +EN t +t GMM (Arellano et al. (1991) (1995), Blundell et al. (1998), Hansen et al. (1996)) Yt 1; EN t Yt= Y t 1+W Y t 1 +EX t +EN t +t LSDV-IV (Korniotis (2008)) Yt 1; EN t Yt= W Y t +EX t +t Spatial-MLE / Spatial 2SLS(Anselin (1988) (2001), Elhorst (2003)) W Y t Minimum Distance (Azomahou (2008)) Yt= W Y t +EX t +EN t +t Spatial 2SLS (Dall erba & Le Gallo (2007)) W Y t; EN t Yt= Y t 1+W Y t +EX t +t Spatial Dynamic MLE (Elhorst (2003b, 2005, 2008)) W Y t; Y t 1 Spatial Dynamic QMLE (Yu, de Jong & Lee (2007) (2008), Lee & Yu (2007)) C2SLSDV (Beenstock & Felsenstein (2007)) Spatial MLE-GMM / Spatial MLE-Spatial Dynamic MLE (Elhorst (2008)) Yt= Y t 1+W Y t +EX t +EN t +t System-GMM (Arellano & Bover (1995), Blundell & Bond (1998)) W Y t; Y t 1 ; EN t 7

8 The spatial 2SLS uses the exogenous variables and their spatially weighted averages (EX t, W t EX t ) as instruments 5. When the number of cross-sections is larger than the period sample, Anselin (1988) suggests to estimate the model using MLE, 2SLS or 3SLS in a spatial seemingly unrelated regression (SUR) framework. More recently, Azomahou (2008) proposes to estimate a spatial panel autoregressive model applying a two-step minimum distance estimator. In the presence of endogenous variables, Dall erba & Le Gallo (2007) suggest to estimate a spatial lag panel model, which includes several endogenous variables but no lagged dependent variable, by applying spatial 2SLS with lower orders of the spatially weighted sum of the exogenous variables as instrument for the spatial autoregressive term 6. In a dynamic context, the estimation of spatial lag panel models is usually based on a ML function. Elhorst (2003a, 2005) proposes to estimate the unconditional loglikelihood function of the reduced form of the model in rst-di erence. While the absence of explanatory variables besides the time and spatial lags leads to an exact likelihood function, this is no longer the case when additional regressors are included. Moreover, when the sample size T is relatively small the initial observations contribute greatly to the overall likelihood. That is why the pre-sample values of the explanatory variables and likelihood function are approximated using the Bhargava & Sargan approximation or the Nerlove & Balestra approximation. More recently, Yu et al. (2008) provide a theoretical analysis on the asymptotic properties of the quasi-maximum likelihood (Spatial Dynamic QML), which relies on the maximization of the concentrated likelihood function of the demeaned model. They show that the limit distribution is not centered around zero and propose a bias-corrected estimator 7. Beside the fact that Yu et al. (2008) do not assume normality, the main di erence with Elhorst s ML estimators lies in the asymptotic structure. Elhorst considers xed T and large N (N! 1), while Yu et al. assume large N and T (N! 1; T! 1). Consequently, the way the individual e ects are taken out di ers: Elhorst considers rst-di erence variables, while Yu et al. 5 In a cross-section setting, Kelejian & Prucha (1998) suggest also additional instruments (Wt 2 EX t, Wt 3 EX t,...). Lee (2003) shows that the estimator proposed by Kelejian & Prucha is not an asymptotically optimal estimator and suggests a three-steps procedure with an alternative instrument for the 1 spatial autoregressive coe cient in the last step (W t I N e e bwt EXtb, where e e b and b are estimates obtained using the S2SLS proposed by Kelejian & Prucha (1998)). 6 Recently, Fingleton & Le Gallo (2008) propose an extended feasible generalized spatial two-stage least squares estimator for spatial lag models with several endogenous variables and spatial error term in a cross-section framework. 7 In two other related working papers, Lee & Yu (2007) and Yu et al.(2007) investigate the presence of non-stationarity and time xed e ects, respectively, in a spatial dynamicpanel framework. 8

9 demean the variables. Assuming large T avoids the problem associated with initial values and the use of approximation procedures. Finally, Yu et al s approach allows to recover the estimated individual e ects, which is not the case with the estimator proposed by Elhorst. In a recent paper, Elhorst (2008) analyzes the nite sample performance of several estimators for a spatial dynamic panel model with only exogenous variables. The estimators considered are the Spatial MLE, Spatial Dynamic MLE and GMM. His Monte Carlo study shows that Spatial Dynamic MLE has the better overall performance in terms of bias reduction and root mean squared errors (RMSE), although the Spatial MLE presents the smallest bias for the spatial autoregressive coe cient. Based on these results, Elhorst proposes two mixed estimators, where the spatial lag dependent variable is based on the spatial ML estimator and the remaining parameters are estimated using either GMM or Spatial Dynamic ML conditional on the spatial ML s estimate of the spatial autoregressive coe cient. These two mixed estimators outperform the original estimators. The mixed Spatial MLE/Spatial Dynamic MLE estimator shows superior performance in terms of bias reduction and RMSE in comparison with mixed Spatial MLE/GMM. However, the latter can be justi ed on a practical ground if the number of cross-sections in the panel is large, since the time needed to compute Spatial MLE/Spatial Dynamic MLE is substantial. In a spatial vector autoregression (VAR) setting, Beenstock & Felsenstein (2007) suggest a two-step procedure. The rst step consists of applying LSDV to the model without the spatial lag and computing the tted values ( Y b t ). Then, in the second step, the full model is also estimated using LSDV, but with W tyt b as instrument for W t Y t. Finally, the authors suggest to correct the bias of the lagged dependent variable by using the asymptotic bias correction de ned by Hsiao(1986). If one is willing to consider some explanatory variables as potentially endogenous in a dynamic spatial panel setting, then no spatial estimator is currently available. In many empirical applications, endogeneity can arise from measurement errors, variables omission or the presence of simultaneous relationship(s) between the dependent and the covariate(s). The main drawback of applying SMLE, SDMLE or SDQMLE is that, while the spatial autoregressive coe cient is considered endogenous, no instrumental treatment is applied to other potential endogenous variables. This can lead to biased estimates, which would invalidate empirical results. This is con rmed by the Monte Carlo analysis performed in the next section. 9

10 2.2 System GMM Empirical papers dealing with a spatial dynamic panel model with several endogenous variables usually apply system-gmm (see for example Madriaga & Poncet (2007), Foucault, Madies & Paty (2008), or Hong, Sun & Li (2008)). Back in 1978, Haining proposed to instrument a rst order spatial autoregressive model using lagged dependent variables. While this method is not e cient in a cross-section setting, because it does not use ef- ciently all the available information (Anselin (1988)), this is no longer necessarily the case in a panel framework. The bias-corrected LSDV-IV estimator proposed by Korniotis (2007) is in line with this approach and considers lagged spatial lag and dependent variable as instruments. Accordingly, the use of system GMM might be justi ed in this trade-o situation, since the spatial lag would be instrumented by lagged values of the dependent variable, spatial autoregressive variable as well as exogenous variables 8. For simplicity, equation (2) is reformulated for a given cross-section i (i = 1; ::; N) at time t (t = 1; ::; T ): Y it = Y it 1 + [W t Y t ] i + EX it + EN it + i + v it (3) According to the GMM procedure, one has to eliminate the individual e ects ( i ) correlated with the covariates and the lagged dependent variable, by rewriting equation(3) in rst di erence for individual i at time t: 4Y it = 4Y it [W t Y t ] i + 4EX it + 4EN it + 4v it (4) Even if the xed e ects (within) estimator cancels the individual xed e cts( i ), the lagged endogenous variable (4Y it 1 ) is still correlated with the idiosyncratic error terms (v it ). Nickell (1981) as well as Anderson & Hsiao (1981) show that the within estimator has a bias measured by O( 1 T ) and is only consistent for large T. Given that this condition is usually not satis ed, the GMM estimator is also biased and inconsistent. Arellano & Bond (1991) propose the following moment conditions associated with equation (4): E (Y i;t 4v it ) = 0; for t = 3; :::; T and 2 t 1 (5) 8 Badinger et al. (2004) recommend to apply system GMM, once the data has been spatially ltered. This approach can be consider only when spatial depence is viewed as a nuisance parameter. 10

11 But the estimation based only on these moment conditions (5) is insu cient, if the strict exogeneity assumption of the covariates (EX it ) has not been veri ed. The explicative variables constitute valid instruments to improve the estimator s e ciency, only when the strict exogeneity assumption is satis ed: E (EX i 4v it ) = 0; for t = 3; :::; T and 1 T (6) However, the GMM estimator based on the moment conditions (5) and (6) can still be inconsistent when < 2 and in presence of inverse causality, i.e. E(EX it v it ) 6= 0. In order to overcome this problem, one can assume that the covariates are weakly exogenous for < t, which means that the moment condition (6) can be rewritten as: E (EX i;t 4v it ) = 0; for t = 3; :::; T and 1 t 1 (7) For the di erent endogenous variables, including the spatial lag, the valid moment conditions are E (EN i;t 4v it ) = 0; for t = 3:::T and 2 t 1 (8) E ([W t Y t ] i 4v it ) = 0; for t = 3:::T and 2 t 1 (9) For small samples, this estimator can still yield biased coe cients. Blundell & Bond (1998) show that the imprecision of this estimator is bigger as the individual e ects are important and as the variables are persistent over time. Lagged levels of the variables are weak instruments for the rst di erences in such case. To overcome this limits, the authors propose the system GMM, which estimate simultaneously equation(3)and equation (4). The extra moment conditions for the extended GMM are thus: E (4Y i;t 1 v it ) = 0; for t = 3; :::; T (10) E (4EX it v it ) = 0; for t = 2; :::; T (11) E (4EN it 1 v it ) = 0; for t = 3; :::; T (12) E (4 [W t 1 Y t 1 ] i v it ) = 0; for t = 3; :::; T (13) The consistency of the SYS-GMM estimator relies on the validity of these moment conditions, which depends on the assumption of absence of serially correlation of the level residuals and the exogeneity of the explanatory variables. Therefore, it is necessary to apply speci cation tests to ensure that these assumptions are justi ed. More generally, one should keep in mind that the estimation of the spatial autoregressive coe cient 11

12 via SYS-GMM, although "potentially" consistent, is usually not the most e cient one. E ciency relies on the "proper" choice of instruments, which is not an easy task to determine. Arellano & Bond suggest two speci cation tests in order to verify the consistency of the GMM estimator. First, the overall validity of the moment conditions is checked by the Sargan/Hansen test. Second, the Arellano-Bond test examines the serial correlation property of the level residuals. Another issue lies in the fact that the instrument count grows as the sample size T rises. A large number of instruments can over t endogenous variables (i.e. fail to correct for endogeneity) and leads to inaccurate estimation of the optimal weight matrix, downward biased two-step standard errors and wrong inference in the Hansen test of instruments validity. Okui (2009) demonstrates that the bias of extended GMM does not result from the total number of instruments, but from the number of instruments for each equation. As pointed out by Roodman (2009), it is advisable to restrict the number of instruments by de ning a maximum number of lags and/or by collapsing the instruments 9. The collapse option consists of combining instruments through addition into subsets. For instance, if the instruments are collapsed, the moment condition (5) becomes: E (Y i;t 4v it ) = 0; for 2 t 1 (14) This modi ed moment condition still imposes the orthogonality of Y i;t and 4v it, but rather to hold for each t and, it is only valid for each. Roodman (2009) shows that collapsed instruments lead to less biased estimates, although the associated standard errorstend to increase. GMM results for the remaining of the paper are, thus, based on collapsed instruments A Monte-Carlo Study In this section, we investigate the nite sample properties of several estimators including Spatial MLE, Spatial Dynamic MLE and Spatial Dynamic QMLE, LSDV, di erence GMM and extended GMM to account for the endogeneity of the spatial lag as well as an additional regressor in a dynamic panel data context using Monte-Carlo simulations This approach has been adopted in several empirical papers, including Beck & Levine (2004). 10 See appendix 6.A for further details on extended GMM and appendix 6.B for spatial ML estimators. 11 All simulations are performed using Matlab R2008b. 12

13 Simulation studies already showed that the bias associated with the spatial lag is rather small (Franzese & Hays (2007), Elhorst (2008)), but none analyzes the consequences of an additional endogenous explanatory variable in a spatial dynamic context. The data generating process (DGP) is de ned as follows: Y it = Y i;t 1 + [W Y t ] i + EX it + EN it + i + v it (15) EX it = EX i;t 1 + u it (16) EN it = EN i;t 1 + i + v it + e it (17) with i N 0; 2 ; vit N 0; 2 v ; uit N 0; 2 u ; eit N 0; 2 e. In order to avoid results being in uenced by initial observations, the covariates Y i0, EX i0 and EN i0 are set to 0 for all i and each variable is generated (100 + T ) times according to their respective DGP. The rst 100 observations are then discarded. Note that the dependent variable is generated according to the reduced form of equation (14): Y it = (1 [W ] i ) 1 [Y i;t 1 + EX it + EN it + i + v it ] (18) Following Kapoor et al. (2007) and Kelejian & Prucha (1999), we consider four di erent types of spatial weight matrix. In each case, the matrix is row-standardized so that all non zero elements in each row sum to one. The matrices considered rely on a perfect "idealized" circular world. More precisely, the three "theoretical" spatial matrices, referred as "1 ahead and 1 behind", "3 ahead and 3 behind" and "5 ahead and 5 behind", respectively, are characterized by di erent degree of sparseness. Each are such that each location is related to the one/three/ ve locations immediately before and after it, so that each nonzero elements are equal to 0:5/0:3/0:1, respectively. In addition, as a robustness check, we consider real data on the distance between capitals among 224 countries 12. In order to avoid giving some positive weight to very remote countries (with weaker cultural, political and economic ties), we consider the negative exponential weighting scheme. This is done by dividing the distance between locations j and k by the minimum distance within the region r (where location j lies within region r): w (d j;k ) = exp ( d j;k =MIN r;j ) if j 6= k. 12 The data is taken from CEPII database: 13

14 The Monte Carlo experiments rely on the following designs: T 2 f10; 20; 30; 40g ; N 2 f20; 30; 50; 70g ; 2 f0:2; 0:4; 0:5; 0:7g ; 2 f0:1; 0:3; 0:5; 0:7g ; = 1; = 0:65; = 0:5; = 0:45; = 0:25; = 0:6; 2 u = 0:05 2 v = 0:05; 2 e = 0:05; In order to ensure stationarity, only designs which respect the restrictions jj < 1! max if 0 or jj < 1! min if < 0 are considered. The total number of designs is restricted to 160. For each of these designs, we performed 1000 trials. Note that for each design, the initial conditions and spatial weight matrices are generated once. As a measure of consistency, we consider the root mean square error (RMSE). Theoretically, RMSE is de ned as the square root of the sum of the variance and the squared bias of the estimator. Therefore, the RMSE assesses the quality of an estimator in terms of its variation and unbiasedness. We also consider the approximated RMSE given in Kelejian & Prucha (1999) and Kapoor et al. (2007), which converges to the standard RMSE under a normal distribution: s IQ RMSE = bias :35 where the bias is the di erence between the true value of the coe cient and the median of the estimated coe cients, and IQ is the di erence between the 75% and 25% quantile. This de nition has the advantage of being more robust to outliers that may be generated by the Monte-Carlo simulations. 2 The Monte Carlo investigation highlights several important facts 13. First, the results are qualitatively similar with respect to di erent spatial weight schemes, that is why for sake of brevity we only present the results for the "1 ahead and 1 behind" weighting matrix. Second, the results in terms of bias and e ciency depend on the values assigned to the spatial and time lag parameters. Since the results based on RMSE and approximated RMSE are qualitatively similar, we only present the results associated with standard RMSE. In order to assess the global properties of the estimators, we rst discuss the results obtained by averaging over the parameters values of the time and spatial lag coe cients.. 13 The full results are given in appendix 6.C and 6.D. 14

15 3.1 Extended GMM vs. Di erence GMM We rst investigate the consistency and e ciency of the di erence and system GMM estimators in a spatial dynamic panel framework. We consider two-step GMM based on a collapsed instruments structure. To avoid imprecise estimate of the optimal weight matrix (when T > N), we restrict the lags to two and three. In other words, each endogenous variables (i.e. Y t 1, W Y t ; EN t ) is instrumented with their 2nd and 3rd lags values (using the collapse option) and the exogenous variables X t. T = 10 T = T = 30 T = N LSDV DIFF GMM SYS GMM Figure 1: Bias LSDV vs. GMM We consider global consistency and e ciency of GMM estimators: the bias and RMSE results are averaged over the whole range of parameters. For illustrative purpose, we compare the performance of GMM estimators with respect to LSDV estimator. As gure 1 and 2 indicate, system and di erence GMM outperform the xed e ect estimator in terms of bias and e ciency. In fact, LSDV estimator has a negative bias which does not decrease with the size of the panel, invalidating LSDV as a consistent and e cient estimator. Consequently, spatial dynamic panel model should de nitively not be estimated with LSDV. The di erences in terms of performance between di erence 15

16 and system GMM are relatively marginal. Hayakawa(2007) shows that di erence and level GMM estimators can be more biased than system GMM, because the bias of extended-gmm is the sum of two elements. The rst one is the weighted sum of the bias associated with di erence and level GMM, while the second one originates from using the level and di erence estimators jointly. Since the bias of di erence and level GMM evolve in opposite direction, the rst component of the bias of system GMM will tend to be small due to a partially cancelling out e ect. In addition, the weight of the rst element plays also a role in the di erence of the magnitudes of the biases. The latter could explain why both GMM estimators performs almost equivalently. However as depicted in gure 2, system GMM seems to be more e cient in small sample. That is why, we decide to focus on extended GMM T = 10 T = 20 T = 30 T = N LSDV DIFF GMM SYS GMM Figure 2: RMSE LSDV vs. GMM 16

17 3.2 Extended GMM vs. Spatial Estimators In this section, we compare the global performance of extended GMM with respect to spatial ML estimators 14. We consider the average value of the bias and RMSE for the di erent values of the autoregressive and spatial autoregressive coe cient. For each parameter, gure 3 plots the averaged bias of the respective estimators over di erent range of cross-sectional dimension, N 15. The upper panels of gure 3 display the average bias associated with the autoregressive and spatial autoregressive variables respectively, while the lower panels present the average bias for the endogenous and exogenous variables. Although the bias associated with the spatial estimators seems to be independent of the cross-section dimension, it does decrease as the time dimension increases. This corroborates the nding in Yu et al., that the spatial estimators remain slightly biased as the number of cross-section increases (see table 2 and 3 in Yu et al. (2008)). Unlike spatial maximum likelihood estimators, extended GMM s bias decreases as N and/or T increase. As displayed by the plot on the top left of gure 3, all estimators tend to underestimate the autoregressive parameter, (e.g. positive bias), although extended GMM exhibits the smallest bias. The opposite happens for the spatial autoregressive parameter. Note that the spatial estimators tend to dominate the extended GMM in relatively small sample. This is not surprising since spatial estimators explicitly account for the spatial structure of the data by estimating the reduced form of the model. Still as the time dimension increases, extended GMM tends to outperform the spatial estimators. The exogenous variable is also overestimated by all the spatial estimators, although its estimation with system GMM clearly dominates the other estimators. Most importantly, the plot on the bottom left of gure 3 shows how important it is to correct for the endogeneity. In fact, if not corrected, the bias associated with the endogenous variable can represent more than 60% of the true value of the parameter, which is unacceptable. Moreover, the magnitude of the bias of the endogenous covariate does not seem to depend on the sample dimension (N and T ). This nding suggests that estimating a spatial dynamic panel model with endogenous variables using traditional spatial estimators would not be advisable. Overall, extended GMM exhibits superior performance in terms of bias. Beside SYS-GMM, spatial dynamic QMLE displays the lower bias for all coe cients. 14 Note that the spatial estimators rely on a numerical optimization which can lead to non-convergent solution. When convergence is not obtained, the trial is dropped from the analysis. See appendix 6.B for further details about the implementation of the spatial estimators. 15 We take average value of the bias across di erent designs of autoregressive and spatial autoregresive parameters 17

18 Time lag bias Spatial lag bias T = 10 T = 20 T = 10 T = 20 T = 30 T = N SMLE SDMLE SDQMLE SYS GMM T = 30 T = N SMLE SDMLE SDQMLE SYS GMM Endogenous bias Exogenous bias T = 10 T = 20 T = 10 T = 20 T = 30 T = N SMLE SDMLE SDQMLE SYS GMM T = 30 T = N SMLE SDMLE SDQMLE SYS GMM Figure 3: Bias Extended GMM vs. Spatial Estimators 18

19 Figure 4 depicts the performance of the estimators in terms of e ciency according to RMSE. The results are slightly di erent from the ones we obtained in the analysis of the bias. Despite the fact that spatial ML estimators yield more bias, they tend to be more e cient than extended GMM in small samples for most parameters, except the endogenous parameter. This is con rmed by gure 5, which plots the histogram distribution for each estimated parameter based on 1000 trials for N = 70 and T = 20. However, as the panel size increases, the performance of GMM converges to the same level of e ciency of the spatial ML estimators. Interestingly, the estimation in moderate samples of the time lag and exogenous variables with SYS-GMM can be even more e cient than with spatial estimators. In other words, extended GMM might be more consistent and e cient than spatial ML estimators. This result might seems to be a paradox with respect to the accepted notion that ML is more e cient than GMM, but it is not. Actually, this nding is an extension to the panel framework of Das et al. s (2003) conclusion that FG2SLS can outperform ML estimators in moderate cross-section samples. The main reason for this seemingly contradictory result is that the spatial ML approach requires the estimation of more parameters than does extended GMM. ML involves the estimation of the additional parameter 2. Moreover, Elhorst s approach implies the estimation of the initial condition restrictions, while Yu et al s method is explicitly designed for large N and T. Therefore, the classical arguments relating to relative e ciency do not apply here. In addition, as gures 3 and 4 indicate, the spatial estimators seem to reach e ciency in relatively small samples (making the slope of RMSE and bias almost at), while extended GMM tends to gain e ciency in moderate sample (the slope of RMSE and bias being more steeper). As we already mentioned it in the bias analysis, the estimation of the endogenous covariate is clearly more e cient with extended GMM than with any of the spatial maximum likelihood estimators. The slope of the RMSE line in gure 4 is almost null for the QMLE and MLE, which suggests that increasing the dimension sample cannot improve e ciency of the estimate of the endogenous variable. In other words, the use of spatial ML estimators is not recommended in the presence of endogenous or predetermined variable. Among the spatial estimators, SDQMLE is again the best one in terms of e ciency. Note, that the estimation of the spatial lag parameter by simple spatial MLE is as e cient as by the other spatial estimators. This observation was highlighted by Elhorst (2008) who suggests to estimate a dynamic panel model with exogenous variables by rst estimating the spatial lag with SMLE and then estimating the remaining parameters using either SDMLE or GMM. 19

20 Time lag RMSE Spatial lag RMSE T = 10 T = 20 T = 10 T = 20 T = 30 T = N LSDV DIFF GMM SYS GMM T = 30 T = N LSDV DIFF GMM SYS GMM Endogenous RMSE Exogenous RMSE T = 10 T = 20 T = 10 T = 20 T = 30 T = N LSDV DIFF GMM SYS GMM T = 30 T = N LSDV DIFF GMM SYS GMM Figure 4: RMSE Extended GMM vs. Spatial Estimators 20

21 Time lag true value = 0.5 Spatial lag true value = SMLE Density Density SDMLE SDQMLE Density Density Density Density SMLE SDMLE Density Density SYS GMM SDQMLE SYS GMM Endogenous variable true value = 0.5 Exogenous variable true value = SMLE Density Density SDMLE SDQMLE Density Density Density Density SMLE SDMLE Density Density SYS GMM SDQMLE SYS GMM Figure 5: Histograms of the parameters (N = 70 and T = 20) 21

22 4 Robustness Check In this section, we investigate the sensitivity of our main results. First, we describe the results in terms of RMSE response functions. Second, we check if the results are sensitive to the choice of the spatial matrix W. Additionally, we analyze what happens when the spatial weight scheme is miss-speci ed. Third, we drop the assumption of Gaussian process for the error terms and individual e ects. Last, we investigate the consequence of dealing with spatially dependent endogenous and exogenous variables RMSE response function As previously commented, the relationship between the performance of the estimators and the model parameters is not necessarily easily determined. That is why, we report the results in terms of response functions. Using the RMSE for each estimator and parameters of the entire set of designs, the following equation is estimated by OLS: p log Ni T i RMSE i = a 1 + a 2 1 W i + a 3 i + a 4 i + a 5 ( i i ) + a 6 W i N i + a 7 W i T i + a 8 1 N i + a 9 1 T i + i where RMSE i is the RMSE of a given parameter obtained using a given estimator in the i-th design. Note that W i corresponds to the value attributed to the construction of the spatial weight matrix and is a measure of the degree of sparseness of the matrix W (e.g. W i 2 (1; 3; 5)). Note that the dependent variable is expressed in logarithm in order to rule out negative predicted RMSE. Therefore, the interpretation of the estimated coe cients is not straightforward (i.e. a negative coe cient implies a small (close to zero) e ect). The RMSE response function estimation results are displayed in table 2. Although some estimations are a ected by multicollinearity, the t of the response functions to the data is relatively good as suggested by R 2. The comments made in the previous section are con rmed by the estimation results. As expected, the main factor that contributes to the e ciency of the GMM estimator is the panel size (N and T ). This is not the case 16 To conserve space, we dont display the results tables, but they remain available upon request. 22

23 Table 2: RMSE Response Functions Time lag: Spatial lag: Endogenous variable: Exogenous variable: SYS-GMM SDMLE SDQMLE SYS-GMM SDMLE SDQMLE SYS-GMM SDMLE SDQMLE SYS-GMM SDMLE SDQMLE 1=W i *** *** *** *** *** *** *** (-0.057) (-0.064) (-0.061) (-0.081) (-0.093) (-0.092) (-0.049) (-0.018) (-0.018) (-0.063) (-0.072) (-0.072) i ** *** *** *** *** *** *** *** *** *** *** (-0.105) (-0.122) (-0.115) (-0.143) (-0.15) (-0.151) (-0.087) (-0.032) (-0.032) (-0.109) (-0.12) (-0.121) i *** *** *** *** *** *** *** *** *** *** *** *** (-0.116) (-0.159) (-0.153) (-0.187) (-0.205) (-0.207) (-0.1) (-0.038) (-0.039) (-0.14) (-0.147) (-0.148) i i 1.437*** 4.837*** 5.041*** *** 5.876*** * 2.471*** 2.380*** (-0.384) (-0.436) (-0.406) (-0.556) (-0.646) (-0.66) (-0.317) (-0.121) (-0.123) (-0.433) (-0.461) (-0.461) Wi=N i 0.706** *** 1.991*** 2.055*** (-0.357) (-0.384) (-0.374) (-0.483) (-0.484) (-0.495) (-0.299) (-0.107) (-0.11) (-0.414) (-0.369) (-0.363) Wi=T i * ** 1.366*** 1.310*** * (-0.166) (-0.188) (-0.177) (-0.24) (-0.294) (-0.297) (-0.137) (-0.048) (-0.049) (-0.186) (-0.201) (-0.195) 1=N i 13.40*** *** *** 13.28*** 4.830*** 3.885** 16.85*** *** *** 15.56*** (-1.317) (-1.482) (-1.41) (-1.731) (-1.657) (-1.691) (-1.079) (-0.38) (-0.391) (-1.402) (-1.46) (-1.444) 1=T i 13.22*** *** *** 12.70*** 1.976** 3.291*** 11.04*** *** *** 9.222*** *** *** (-0.613) (-0.723) (-0.669) (-0.865) (-0.92) (-0.939) (-0.52) (-0.166) (-0.173) (-0.647) (-0.723) (-0.711) Constant *** 0.202*** 0.302*** *** *** *** *** 2.476*** 2.474*** *** *** *** (-0.057) (-0.072) (-0.068) (-0.084) (-0.09) (-0.091) (-0.049) (-0.02) (-0.019) (-0.065) (-0.072) (-0.072) R The standard deviation is reported in parenthesis. ** p<0.01, ** p<0.05, * p<

24 for the spatial estimators, because the rate of e ciency is only marginally a ected by the panel size. This was already con rmed by the gures 3 and 4, where the slope of the spatial estimators RMSE were relatively at. The e ects of the autoregressive and spatial autoregressive parameters on the RMSE are clearly non-linear. While the individual e ects of the parameters are negative and signi cant, the interaction term enters the RMSE regression positively and signi cantly in almost all cases. This result means that it is the combination of the time and spatial lag coe cients which a ects the e ciency of the estimators rather than each parameter taken individually. Another important nding relates to the speci cation of the spatial weight matrix W. The performances of the spatial estimators are de nitively more sensitive to the spatial weight matrix than the extended GMM estimator. The spatial lag parameter is the only parameter where performance of GMM seemed to be a ected by the spatial matrix W. This nding suggests that spatial estimators are more sensitive to the speci cation of the spatial weight matrix than GMM. We address this question in the following analysis. 4.2 Role of the Spatial Weight Matrix In order to check the sensitivity of our results to the choice of spatial matrix, we re-run Monte Carlo experiments for three types of spatial matrices: "3 ahead and 3 behind", "5 ahead and 5 behind" and a negative exponential matrix based on real data distance for 224 countries 17. As in the baseline case, we perform 1000 simulations for each type of spatial matrix. The results seems to be qualitatively similar to what we obtained for our baseline case. Extended GMM outperforms the spatial estimators according to unbiasness criteria. The only exception is the spatial lag parameter which is better estimated by spatial ML estimators in relatively small samples. As before, all estimators tend to underestimate autoregressive parameter and overestimate the spatial autoregressive parameter. The coe cients of the endogenous and exogenous variable tend to be overestimated by spatial ML estimators. As before the bias of the endogenous and exogenous variables do not seem to decrease with an increase in the sample size (N and/or T ). As highlighted by the response function analysis, the bias of the spatial estimators seems to increase with the number of non-zeros elements in the spatial weight matrix. This is not the case for SYS-GMM, whose performance seems to be not a ected by the degree of sparseness of the matrix W. Note that the spatial estimators sensitivity to the type of spatial weight matrix tends to disappear as the panel dimension increases. 17 See section 3 for further details. 24

25 Most of the spatial econometrics literature presumes that the spatial weight matrix is known and well speci ed. However, in practice, one has to de ne the spatial weight matrix. This is usually done based on some underlying theory. But since there is no formal theoretical guidance on the choice of the matrix W, the latter can be misspeci ed. In order to study the consequences of the misspeci cation of the spatial weight matrix, we estimate the spatial dynamic panel model using the data generated by the "3 ahead and 3 behind" spatial matrix, but assuming the spatial weight is "1 ahead and 1 behind" and "5 ahead and 5 behind". In the rst case, we assume the spatial dependence to be more local than it actually is while in the second scenario we presume the opposite. The results depend on the type of misspeci cation. In the case of over-spatial dependence, the results are similar to what we obtained in the baseline case, although the parameters bias tends to be higher, this is particular true for system GMM. It seems that assuming more global form of spatial dependence introduces additional noise, which a ects the performance of extended GMM. As found earlier, the autoregressive variable tends to be underestimated while the spatial autoregressive variable, endogenous and exogenous variables tend to be overestimated. Moreover, the estimation of the autoregressive and endogenous variables by the spatial estimators are dominated by SYS-GMM according to the unbiasness criterion. In the case of under-spatial dependence, the results for all estimators are severely a ected. The bias for the spatial autoregressive parameter changes from negative to positive. In fact, limiting spatial dependence to the close neigbourhood tends to underestimate the spatial e ect, although it remains the only parameter which is less a ected. The main explanation for this nding lies in the fact that spatial methods estimate the reduced form of the model, which means that the entire set of parameters, beside the spatial lag, are a ected by the miss-speci cation (see equation (18)). As noted previously, simple SMLE seems to estimate the spatial lag as accurate as SDMLE and SDQMLE (Elhorst (2008)). From an applied econometric point of view, these results suggest that spatial dynamic panel models should not be estimated using only one type of spatial weight matrix. In fact, di erent types of spatial weight matrices (contiguity, geographical distance, economic distance) should be applied to check the robustness and consistency of the results. 25